Question 10: Suppose the temperature (T) at a point (x, y, z) in \mat…

Suppose the temperature ($T$) at a point $(x, y, z)$ in $\mathbb{R}^3$ is given by the function

$T(x, y, z) = e^{x+y+z}$

Let $T_1$, $T_2$, and $T_3$ be the tangent planes to the function $f(x, y) = \sqrt{9 - x^2 - y^2}$ at the points $(1, 2)$, $(2, 1)$, and $(2, 2)$, respectively. The set of points on each plane $T_1$, $T_2$, and $T_3$ form affine subspaces $A_1$, $A_2$, and $A_3$, respectively, of the vector space $\mathbb{R}^3$, with respect to usual addition and scalar multiplication. Let $V_1$, $V_2$, and $V_3$ denote the vector subspaces corresponding to the affine subspaces $A_1$, $A_2$, and $A_3$, respectively.

Answer the subquestions using the given information.